Petr Siegl
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Research Interests
Petr Siegl has a total of 25 co-authors. Here are the top co-authors by number of shared publications:
- Boris Mityagin
Department of Mathematics, The Ohio State University, Columbus, USA
6 shared publications - Antonio Arnal
Institut für Angewandte Mathematik, Technische Universität Graz, Graz, Austria
5 shared publications - Joe Viola
Laboratoire de Mathématiques Jean Leray, LMJL, Nantes Université, Nantes, France
4 shared publications - David Krejcirik
Department of Theoretical Physics, Nuclear Physics Institute Czech Academy of Sciences Řež Czech Republic
3 shared publications - Jussi Behrndt
Institut für Angewandte Mathematik, Technische Universität Graz, Graz, Austria
3 shared publications - David Krejcirik
Department of Theoretical Physics Nuclear Physics Institute ASCR 25068 Řež Czech Republic
2 shared publications
Latest Publications
- Open Access
Spectral projections of an anharmonic oscillator with complex polynomial potential
For a broad class of polynomial potentials $V$, with an important and instructive representative being $V(x) = x^{2a} + i x^b$, $x \in \mathbb R$, $a, b \in \mathbb N$, we show that the system of spectral projections $\{P_n\}_n$ of an anharmonic operator $L = - (\mathrm{d}/ \mathrm{d}x)^2 + V(x)$ does not generate a (Riesz) basis in $L^2(\mathbb R)$ if $a - 1 < b < 2a$. Moreover, for $σ= [b - (a - 1)]/(1 + a)$ and $γ> 0$ small enough, $\limsup_n \|P_n\|/ \exp(γn^σ) = \infty$. Proofs are based on two groups of results which are of great interest on their own: (a) relationship between behavior (growth) of the norms of projections $\|P_n\|$ and of the resolvent $\|(z - L)^{-1}\|$ outside of the spectrum $σ(L)$; (b) partial fraction decompositions of special meromorphic functions $1/F$ where $F(w) = \prod_{k=1}^\infty \left( 1 + \frac{w}{a_k} \right)$, $a_{k+1} \geq a_k>0$, $k \in \mathbb N$, and the generalization of the first resolvent identity.Open AccessSpectral projections of an anharmonic oscillator with complex polynomial potential
For a broad class of polynomial potentials $V$, with an important and instructive representative being $V(x) = x^{2a} + i x^b$, $x \in \mathbb R$, $a, b \in \mathbb N$, we show that the system of spectral projections $\{P_n\}_n$ of an anharmonic operator $L = - (\mathrm{d}/ \mathrm{d}x)^2 + V(x)$ does not generate a (Riesz) basis in $L^2(\mathbb R)$ if $a - 1 < b < 2a$. Moreover, for $σ= [b - (a - 1)]/(1 + a)$ and $γ> 0$ small enough, $\limsup_n \|P_n\|/ \exp(γn^σ) = \infty$. Proofs are based on two groups of results which are of great interest on their own: (a) relationship between behavior (growth) of the norms of projections $\|P_n\|$ and of the resolvent $\|(z - L)^{-1}\|$ outside of the spectrum $σ(L)$; (b) partial fraction decompositions of special meromorphic functions $1/F$ where $F(w) = \prod_{k=1}^\infty \left( 1 + \frac{w}{a_k} \right)$, $a_{k+1} \geq a_k>0$, $k \in \mathbb N$, and the generalization of the first resolvent identity. - Open Access
Weak coupling for Schrödinger operators with complex potentials
We study the discrete eigenvalues emerging from the threshold of the essential spectrum of one or two-dimensional Schrödinger operators with complex-valued $ L^p $-potentials in a weak coupling regime. We derive necessary and sufficient conditions on the potential for the existence or absence of discrete eigenvalues in this regime and also analyze their uniqueness and algebraic multiplicity. Our results can be viewed as natural non-self-adjoint extensions of the well-known classical weak coupling phenomenon for self-adjoint Schrödinger operators with real-valued potentials going back half a century to Simon's famous paper [Simon 1976].
Open AccessWeak coupling for Schrödinger operators with complex potentials
We study the discrete eigenvalues emerging from the threshold of the essential spectrum of one or two-dimensional Schrödinger operators with complex-valued $ L^p $-potentials in a weak coupling regime. We derive necessary and sufficient conditions on the potential for the existence or absence of discrete eigenvalues in this regime and also analyze their uniqueness and algebraic multiplicity. Our results can be viewed as natural non-self-adjoint extensions of the well-known classical weak coupling phenomenon for self-adjoint Schrödinger operators with real-valued potentials going back half a century to Simon's famous paper [Simon 1976].
- Open Access
- 26 November 2025
- Published by Springer Nature in Complex Analysis and Operator Theory
- Vol. 20 (1) , 11
Generalized Boundary Triples for Adjoint Pairs with Applications to Non-Self-Adjoint Schrödinger Operators
We extend the notion of generalized boundary triples and their Weyl functions from extension theory of symmetric operators to adjoint pairs of operators, and we provide criteria on the boundary parameters to induce closed operators with a nonempty resolvent set. The abstract results are applied to Schrödinger operators with complex $$L^p$$ -potentials on bounded and unbounded Lipschitz domains with compact boundaries.- 26 November 2025
- Published by Springer Nature in Complex Analysis and Operator Theory
- Vol. 20 (1) , 11
Open AccessGeneralized Boundary Triples for Adjoint Pairs with Applications to Non-Self-Adjoint Schrödinger Operators
We extend the notion of generalized boundary triples and their Weyl functions from extension theory of symmetric operators to adjoint pairs of operators, and we provide criteria on the boundary parameters to induce closed operators with a nonempty resolvent set. The abstract results are applied to Schrödinger operators with complex $$L^p$$ -potentials on bounded and unbounded Lipschitz domains with compact boundaries. - Open Access
- 21 August 2025
- Published by Springer Nature in Boletín de la Sociedad Matemática Mexicana
- Vol. 31 (3) , 1-36
Local form subordination without a power decay and a criterion of Riesz basesness
We revisit the local form subordination condition on the perturbation of a self-adjoint operators with compact resolvent, which is used to show the Riesz basis property of the eigensystem of the perturbed operator. Our new assumptions and new proof allow for establishing the Riesz basis property also in the case of slow and non-monotone decay in this subordination condition.
- 21 August 2025
- Published by Springer Nature in Boletín de la Sociedad Matemática Mexicana
- Vol. 31 (3) , 1-36
Open AccessLocal form subordination without a power decay and a criterion of Riesz basesness
We revisit the local form subordination condition on the perturbation of a self-adjoint operators with compact resolvent, which is used to show the Riesz basis property of the eigensystem of the perturbed operator. Our new assumptions and new proof allow for establishing the Riesz basis property also in the case of slow and non-monotone decay in this subordination condition.
- 1 August 2025
- Published by Cambridge University Press (CUP) in Proceedings of the Royal Society of Edinburgh: Section A Mathematics
- p. 1-31
Generalized Airy operators
We study the behaviour of the norm of the resolvent for non-self-adjoint operators of the form $A := -\partial_x + W(x)$ , with $W(x) \ge 0$ , defined in ${L^2}({\mathbb{R}})$ . We provide a sharp estimate for the norm of its resolvent operator, $\| (A - \lambda)^{-1} \|$ , as the spectral parameter diverges $(\lambda \to +\infty)$ . Furthermore, we describe the C0-semigroup generated by −A and determine its norm. Finally, we discuss the applications of the results to the asymptotic description of pseudospectra of Schrödinger and damped wave operators, and also the optimality of abstract resolvent bounds based on Carleman-type estimates.- 1 August 2025
- Published by Cambridge University Press (CUP) in Proceedings of the Royal Society of Edinburgh: Section A Mathematics
- p. 1-31
Generalized Airy operators
We study the behaviour of the norm of the resolvent for non-self-adjoint operators of the form $A := -\partial_x + W(x)$ , with $W(x) \ge 0$ , defined in ${L^2}({\mathbb{R}})$ . We provide a sharp estimate for the norm of its resolvent operator, $\| (A - \lambda)^{-1} \|$ , as the spectral parameter diverges $(\lambda \to +\infty)$ . Furthermore, we describe the C0-semigroup generated by −A and determine its norm. Finally, we discuss the applications of the results to the asymptotic description of pseudospectra of Schrödinger and damped wave operators, and also the optimality of abstract resolvent bounds based on Carleman-type estimates.- Open Access
Generalized boundary triples for adjoint pairs with applications to non-self-adjoint Schrödinger operators
We extend the notion of generalized boundary triples and their Weyl functions from extension theory of symmetric operators to adjoint pairs of operators, and we provide criteria on the boundary parameters to induce closed operators with a nonempty resolvent set. The abstract results are applied to Schrödinger operators with complex $L^p$-potentials on bounded and unbounded Lipschitz domains with compact boundaries.
Open AccessGeneralized boundary triples for adjoint pairs with applications to non-self-adjoint Schrödinger operators
We extend the notion of generalized boundary triples and their Weyl functions from extension theory of symmetric operators to adjoint pairs of operators, and we provide criteria on the boundary parameters to induce closed operators with a nonempty resolvent set. The abstract results are applied to Schrödinger operators with complex $L^p$-potentials on bounded and unbounded Lipschitz domains with compact boundaries.
- Open Access
Generalised Airy Operators
We study the behaviour of the norm of the resolvent for non-self-adjoint operators of the form $A := -\partial_x + W(x)$, with $W(x) \ge 0$, defined in $L^2(\mathbb{R})$. We provide a sharp estimate for the norm of its resolvent operator, $\| (A - \lambda)^{-1} \|$, as the spectral parameter diverges $(\lambda \to +\infty)$. Furthermore, we describe the $C_0$-semigroup generated by $-A$ and determine its norm. Finally, we discuss the applications of the results to the asymptotic description of pseudospectra of Schr\"odinger and damped wave operators and also the optimality of abstract resolvent bounds based on Carleman-type estimates.Open AccessGeneralised Airy Operators
We study the behaviour of the norm of the resolvent for non-self-adjoint operators of the form $A := -\partial_x + W(x)$, with $W(x) \ge 0$, defined in $L^2(\mathbb{R})$. We provide a sharp estimate for the norm of its resolvent operator, $\| (A - \lambda)^{-1} \|$, as the spectral parameter diverges $(\lambda \to +\infty)$. Furthermore, we describe the $C_0$-semigroup generated by $-A$ and determine its norm. Finally, we discuss the applications of the results to the asymptotic description of pseudospectra of Schr\"odinger and damped wave operators and also the optimality of abstract resolvent bounds based on Carleman-type estimates. - Open Access
Resolvent estimates for one-dimensional Schrödinger operators with complex potentials
We study one-dimensional Schr\"odinger operators $\operatorname{H} = -\partial_x^2 + V$ with unbounded complex potentials $V$ and derive asymptotic estimates for the norm of the resolvent, $\Psi(\lambda) := \| (\operatorname{H} - \lambda)^{-1} \|$, as $|\lambda| \to +\infty$, separately considering $\lambda \in \operatorname{Ran} V$ and $\lambda \in \mathbb{R}_+$. In each case, our analysis yields an exact leading order term and an explicit remainder for $\Psi(\lambda)$ and we show these estimates to be optimal. We also discuss several extensions of the main results, their interrelation with some aspects of semigroup theory and illustrate them with examples.Open AccessResolvent estimates for one-dimensional Schrödinger operators with complex potentials
We study one-dimensional Schr\"odinger operators $\operatorname{H} = -\partial_x^2 + V$ with unbounded complex potentials $V$ and derive asymptotic estimates for the norm of the resolvent, $\Psi(\lambda) := \| (\operatorname{H} - \lambda)^{-1} \|$, as $|\lambda| \to +\infty$, separately considering $\lambda \in \operatorname{Ran} V$ and $\lambda \in \mathbb{R}_+$. In each case, our analysis yields an exact leading order term and an explicit remainder for $\Psi(\lambda)$ and we show these estimates to be optimal. We also discuss several extensions of the main results, their interrelation with some aspects of semigroup theory and illustrate them with examples. - 31 August 2022
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Mathematical Analysis
- Vol. 54 (4) , 5064-5101
Diverging Eigenvalues in Domain Truncations of Schrödinger Operators with Complex Potentials
Diverging eigenvalues in domain truncations of Schrödinger operators with complex potentials are analyzed and their asymptotic formulas are obtained. Our approach also yields asymptotic formulas fo...
- 31 August 2022
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Mathematical Analysis
- Vol. 54 (4) , 5064-5101
Diverging Eigenvalues in Domain Truncations of Schrödinger Operators with Complex Potentials
Diverging eigenvalues in domain truncations of Schrödinger operators with complex potentials are analyzed and their asymptotic formulas are obtained. Our approach also yields asymptotic formulas fo...
- Open Access
- 2 August 2022
- Published by Springer Nature in Journal of Fourier Analysis and Applications
- Vol. 28 (4) , 1-34
Concentration of Eigenfunctions of Schrödinger Operators
We consider the limit measures induced by the rescaled eigenfunctions of Schrödinger operators with even confining potentials. We show that the limit measure is supported on $$[-1,1]$$ and with the density proportional to $$(1-|x|^\beta )^{-1/2}$$ when the non-perturbed potential resembles $$|x|^\beta $$ , $$\beta >0$$ , for large x, and with the uniform density for super-polynomially growing potentials. We compare these results to analogous results in orthogonal polynomials and semiclassical defect measures.- 2 August 2022
- Published by Springer Nature in Journal of Fourier Analysis and Applications
- Vol. 28 (4) , 1-34
Open AccessConcentration of Eigenfunctions of Schrödinger Operators
We consider the limit measures induced by the rescaled eigenfunctions of Schrödinger operators with even confining potentials. We show that the limit measure is supported on $$[-1,1]$$ and with the density proportional to $$(1-|x|^\beta )^{-1/2}$$ when the non-perturbed potential resembles $$|x|^\beta $$ , $$\beta >0$$ , for large x, and with the uniform density for super-polynomially growing potentials. We compare these results to analogous results in orthogonal polynomials and semiclassical defect measures.